Remark 1 
Suppose the discretization of the linear operator \(\mathbb {T}_k\)  in Lemma REF  is \(D_k-E_k-E_k^*\)  where \(D_k\)  is the diagonal part, \(-E_k\)  represents the strict lower triangular part and \(E_k^*\)  is the transpose of \(E_k\) . Here we still use \(\mathbb {T}_k\)  as its corresponding discrete matrix. If choosing \( \mathbb {M}_k\)  as the symmetric Gauss-Seidel preconditioner involving \( \mathbb {T}_k\) , considering the positive definiteness requirement of \(M_k\)  as in (REF ), we can choose [1]} (chapter 4.1) (or [2]})
\(\mathbb {M}_k = \mathbb {T}_k + E_k^*D_k^{-1}E_k + \eta I = D_k-E_k-E_k^*+ E_k^*D_k^{-1}E_k + \eta I,\) 
